Journal of Inequalities in Pure and Applied Mathematics

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Journal of Inequalities in Pure and Applied Mathematics

http://jipam.vu.edu.au/

Volume 7, Issue 1, Article 3, 2006

A CONJECTURE ON GENERAL MEANS

OLIVIER DE LA GRANDVILLE AND ROBERT M. SOLOW DEPARTMENT OFECONOMICS

UNIVERSITY OFGENEVA,

40 BOULEVARD DUPONT D’ARVE, CH-1211 GENEVA4 SWITZERLAND.

[email protected] DEPARTMENT OFECONOMICS

MASSACHUSETTSINSTITUTE OFTECHNOLOGY

77 MASSACHUSETTSAVENUE

CAMBRIDGE, MA 02139-4307 [email protected]

Received 20 December, 2005; accepted 16 January, 2006 Communicated by A.M. Rubinov

ABSTRACT. We conjecture that the general mean of two positive numbers, as a function of its order, has one and only one inflection point. No analytic proof seems available due to the extreme complexity of the second derivative of the function. We show the importance of this conjecture in today’s economies.

Key words and phrases: General means, Inflection point, Production functions.

2000 Mathematics Subject Classification. 26E60.

Let x₁, . . . , x_n ben positive numbers andM(p) = (Pn

i=1f_ix^p_i)¹^p the mean of order p of the x_i’s;0< f_i <1andPn

i=1f_i = 1. One of the most important theorems about a general mean is that it is an increasing function of its order. A proof can be found in [2, Theorem 16, pp. 26-27].

The proof rests in part on Hölder’s inequality and on successive contributions to the theory of inequalities that go back to Halley and Newton. Another, more analytic proof is based upon the convexity ofxlogx– see [3, pp. 76-77].

In this note, we make a conjecture about the exact shape of the curveM(p)in(M, p)space.

If it is well known thatM(p)is increasing withp, it seems that the exact properties of the curve M(p)have not yet been uncovered. We offer here a conjecture and explain its importance.

In(M, p)space the curveM(p)has one and only one inflection point ifn = 2, irrespective of the size of thex_i’s and thef_i’s. Between its limiting values,M(p)is in a first phase convex and then turns concave. Due to the extreme complexity of the second derivative ofM(p), we

ISSN (electronic): 1443-5756

370-05

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2 OLIVIER DELAGRANDVILLE ANDROBERTM. SOLOW

could not offer an analytical proof of this property, and we had to rely on numerical calculations only.

We would have liked to extend this conjecture to n > 2, but Professor Anthony Pakes of Western Australia University mentioned to us that his colleague, Grant Keady, found a counter- example with x_i = 1/8,2/9,1 and f_i = 1/27,25/27,1/27 where the second derivative, al- though showing very little variation over[−10,+10],has three zeros. We keep the conjecture forn = 2,whose analytical proof remains, in our opinion, a formidable challenge.

The importance of this property stems from the following reason. Both theory and empirical observations have led economists to introduce and make the widest use of a general mean of orderpin the following form. Letx1 ≡Ktdenote the stock of capital of a nation at time datet;

letx₂ ≡L_tbe the quantity of labour; associated to both variablesK_tandL_tis a function which gives outputY_tas the general mean

(1) Yt= [δK_t^p+ (1−δ)L^p_t]¹^p whereδand(1−δ)are the weights ofKtandLtrespectively.

Furthermore, the orderpof this mean is related to a parameter of fundamental importance, the so-called “elasticity of substitution”, defined as follows. Omitting the time indexes in our notation, let Y denote a given level of production. The equation of the level curve, in space (K, L), corresponding to a given valueY is given, in implicit form, by:

(2) Y = [δK^p+ (1−δ)L^p]¹^p

This level curve is called an "isoquant".

Let the capital-labour ratioK/L≡r, and minus the slope of the level curve be denotedτ:

(3) − dK

dL _Y_{= ¯}_Y

= ∂Y /∂L

∂Y /∂K =τ

The elasticity of substitution, denotedσ, is the elasticity ofrwith respect toτ, defined by

(4) σ = dlogr

dlogτ = dr/r dτ /τ.

From a geometric point of view, the elasticity of substitution measures, in linear approximation, the relative change along an isoquant of the ratior=K/Linduced by a relative change in the slope of the isoquant¹.

It can be verified from (2), (3), (4) that the order of the meanpis related toσbyp= 1−1/σ.

Observations show that σ is close to one, i.e. thatpis close to 0. In turn this implies that the meanY is close to its limiting form when p → 0, the geometric meanY = K^δL^1−δ. It turns out also that the abscissa of the inflection point, for the usual values ofδ(its order of magnitude is 0.3) is very close top = 0. This means that ifσ changes – and we have evidence that it has been increasing in recent years – it has a very significant impact on the production (and income) of an economy.

Note also that not only Y is a general mean of order one, but so is a variable of central importance, income per person, denotedy ≡ Y /L. Indeed, dividing both sides of (1) by Lwe have (dropping the subscript):

y=Y /L= [δr^p+ (1−δ)]¹^p,

a general mean ofr and 1 of orderp. The above conjecture gives the mathematical reason of the considerable impact that a change in the elasticity of substitution in any given economy may have both on income per person and its growth rate.

1The derivation of the production function can be found in [1]

J. Inequal. Pure and Appl. Math., 7(1) Art. 3, 2006 http://jipam.vu.edu.au/

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A CONJECTURE ONGENERALMEANS 3

REFERENCES

[1] K. ARROW, H. CHENERY, B. MINHASANDR. SOLOW, Capital-labor substitution and economic efficiency, The Review of Economics and Statistics, 43(3) (1961), 225–250.

[2] G. HARDY, J.E. LITTLEWOODANDG. PÓLYA, Inequalities, Second Edition, Cambridge Mathe- matical Library, Cambridge, 1952.

[3] D.S. MITRINOVI ´C, Analytic Inequalities, Springer-Verlag, New York, 1970.

J. Inequal. Pure and Appl. Math., 7(1) Art. 3, 2006 http://jipam.vu.edu.au/